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The 1[ratio ]2 mode interaction in exactly counter-rotating von Kármán swirling flow

Published online by Cambridge University Press:  28 March 2003

C. NORE
Affiliation:
Université Paris XI, Département de Physique, 91405 Orsay cedex, France Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur, CNRS, BP 133, 91403 Orsay cedex, France
L. S. TUCKERMAN
Affiliation:
Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur, CNRS, BP 133, 91403 Orsay cedex, France
O. DAUBE
Affiliation:
Université d'Evry Val d'Essonne, 40 rue du Pelvoux, 91020 Evry cedex, France
S. XIN
Affiliation:
Université Paris XI, Département de Physique, 91405 Orsay cedex, France Laboratoire d'Informatique pour la Mécanique et les Sciences de l'Ingénieur, CNRS, BP 133, 91403 Orsay cedex, France

Abstract

The flow produced in an enclosed cylinder of height-to-radius ratio of two by the counter-rotation of the top and bottom disks is numerically investigated. When the Reynolds number based on cylinder radius and disk rotation is increased, the axisymmetric basic state loses stability and different complex flows appear successively: steady states with an azimuthal wavenumber of 1; travelling waves; near-heteroclinic cycles; and steady states with an azimuthal wavenumber of 2. This scenario is understood in a dynamical system context as being due to a nearly codimension-two bifurcation in the presence of $O(2)$ symmetry. A bifurcation diagram is determined, together with the most dangerous eigenvalues as functions of the Reynolds number. Two distinct types of near-heteroclinic cycles are observed, with either two or four bursts per cycle. The physical mechanism for the primary instability could be the Kelvin–Helmholtz instability of the equatorial azimuthal shear layer of the basic state.

Type
Research Article
Copyright
© 2003 Cambridge University Press

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